Stability analysis of multiple-block sliding surfaces

Int..l. Rock Mech. Min. Sci. & Geomech. Ahstr. Vol.30, No.7, pp. 1579-1584, 1993

0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd

Printed in Great Britain

Stability Analysis of Multiple-Block Sliding Surfaces C.F. LEUNG'I" K. W. LOt A relatively simple vectorial method has been developed to tackle polygonal sliding surfaces involving two or more free blocks. Goodman's concept of "twoblock sliding" mechanism has been adopted whereby a transfer of an interaction force is activated from an upper unstable active block to an adjacent lower stable passive block. Furthermore, Goodman's graphical approach has been automated by the vectorial method and extended to handle multiple-block sliding surfaces. The entire analysis may be carried out on a micro-computer and has been proven to be applicable to a fieM case study involving slope excavations in weathered sedimentary rock.

INTRODUCTION Rock slopes may be kinematically unstable where there are weakness planes within the rock mass. Basic modes of rock slope failure include plane sliding, wedge sliding and toppling [1]. Complex, jointed and bedded rock masses may be composed of several blocks interacting with one another along internal weakness planes resulting in more complicated modes of failure. In such cases, plane sliding, wedge sliding and toppling may occur simultaneously or progressively. A block may also slide along polygonal sliding surfaces [2] involving two or more blocks. The stability analysis of multiple-block sliding surfaces is usually tackled by means of the rather simplistic stereographic projection technique [3] or more elaborate Block Theory [4]. In this paper, a relatively simple vectorial method is presented to fully automate the analysis on a micro-computer. The approach is an extension of previous work [5, 6] on the stability of rock excavations involving single sliding blocks.

Internal . shear ~" plane12~. /

Block I. .

W1 ]External I C /sliding

..~ -I~; ! VI plane7

Plane2J--~_l~3 /

/

W3

.~ _~/" \

~ ~

/ /112 "~ ~'~/

~I

shdmg

plane 2

sliding IN3 plane3

Symbols:

N K R I

Normal of weakness plane Reaction force Net resultant force of individual block Interaction force between blocks Friction angle C Cohesion W Serf weight of block (Forces due to water pressure, earthquake and rock bolts etc are omitted for clarity)

Fig. 1. Forces aeting on multiple-block sliding surfaces

STABILITY ANALYSIS The analytical concept generally follows Goodman's [3] approach of a "two-block sliding" mechanism involving active and passive sliding blocks. In the present work, a vectorial approach has been adopted to replace Goodman's graphical approach and to extend the analysis for multiple-block sliding surfaces. The forces acting on a sample polygonal sliding surface consisting of three free blocks are shown in Figure 1. The numbering of the blocks is such that the first block (i= I) is nearer to the slope crest while the last block (i=n) is nearer to the slope base.

tDepartment of Civil Engineering, National University of Singapore, Kent Ridge, Singapore 0511. 1579

The weakness planes at the base of each block are termed as the external sliding surfaces (for example sliding plane 1 for block 1) while the weakness planes between adjacent blocks are termed as the internal shear surfaces (for example shear plane 12 for plane between blocks 1 and 2). Suppose D 1 is the vector representing the resultant of all the disturbing forces acting on block 1 given by

D l = W l + E l +U1 + Vx

(1)

where W 1 is the vector due to self weight of block which is acting vertically downwards, E l is the vector of external force due to earthquake force, Ul is the force vector due to uplift water force which acts normally to the sliding surface and is determined by the position of the

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R O C K M E C H A N I C S IN THE t 9 9 0 s

water table, and V1 is the vector of water force in tension cracks. Since the magnitudes and direction of all the disturbing forces can be readily determined, Equation (1) can be computed by vector algebra such that DI = d~ti + clyd + d, lk

(2)

where i, j and k are the unit vectors in the x, y and z directions, respectively. Using the sign convention whereby downwards is positive, the magnitude dr, trend cqt, and plunge 3dr, of vector D; are given by dI

----v/(dxl2 + dyl2 + dzt 2)

(3)

cqt = tan't(dxJdyl)

(4)

~., = tan-'{ Ida, I # ( ~ . ? + d~?)}

(5)

The quadrant where cqt lies is governed by the sign of dxl and dyt while the magnitude of Bdt should always be between 0° and 90°. The net resultant force Rt of disturbing and resisting forces acting on block 1, ignoring friction and ignoring block 2, may then be given by RI = Dl + Cml + BI = rx~i + ry~y + r~,z

(6)

where Cml is the vector denoting the mobilised shear force and is given by CI/F, Ct is the resisting force contribution by the cohesion acting upwards along plane 1, and F is the factor of safety, and B~ is the vector denoting the resultant of external resisting forces acting on block 1 contributed by rock bolts and shotcrete etc. The magnitude and orientation of vector R1 may be determined in a similar fashion by Equations (3), (4) and (5). According to the friction cone concept [7], an individual block would be stable and held by friction of the sliding plane if the vector representing the net resultant force falls inside the friction cone of the external sliding surface of the block. Let NI be the vector representing the upward normal of plane 1 having dip direction c~t and dip angle BI, which is given by Nt = (sin oq sin 31)i + (cos cq sin 31)J + (-cos #,)k

(7)

If N l denotes the unit vector for N t and the angle of friction for plane 1 is ~ , block 1 would be an active and unstable block if the resultant falls outside the friction cone of plane 1 such that

Nt.R~ > cos ¢ml

(8)

where R 1 denotes the unit vector for R~ and Om~ is the mobilized angle of friction and is given by tan'l(tan~JF). On the other hand, block 1 would be a passive and stable block ifN~.Rt g cos ~,~ indicating that the resultant force falls inside the friction cone.

For an active block, there will be a net transfer of interaction force from block 1 to the next lower block 2. Following Goodman's approach [3], the force is assumed to act at an angle of friction of qt~2 away from the n o ~ l of shear plane 12. The interaction force is denoted by vector I12 given by I12 = KI - Rl

(9)

where K t is the reaction force for block 1 which acts at an angle ~b~ away from the normal of sliding plane 1 for the condition of equilibrium (see Figure 1). In the case of the passive block, no transfer of supporting interaction force is assumed to take place and hence the interaction force is taken to be zero. A similar method of analysis may be conducted for block 2 in addition to which an equal and opposite interaction force 112 has to be taken into account. The net resultant force vector R 2 is given by R 2 = D z + C ~ + B 2 -112

(lo)

where all vector symbols bear the same meaning as before except that subscript 2 now refers to block 2. The stability analysis of block 2 may thereby be analyzed on the basis of the friction cone concept and there will be a transfer of interaction force, Iz3, along shear plane 23 to the next lower block 3 in the manner described earlier for block 1. The stability analysis is hence conducted in a sequential manner for all blocks ranging from the uppermost block 1 until the lowest block n. The entire system of blocks is deemed to be stable if the resultant force of the lowest block lies within the friction cone :of the sliding surface, In Figure 1, for example, where the lowest block is numbered 3, the entire system of blocks would be stable, with passive blocks supporting the active blocks of the system, provided that N3.R 3 ~ cos 0m3

(11)

(Again, vector symbols bear the same meaning as before, with subscript 3 referring to block 3,) In the analysis, the value of factor of safety may be taken as one, initially, in order to evaluate the stability of the entire system. If the system is subsequently found to be stable, the factor of safety against sliding failure may then be evaluated by iteratively by trial and error such that the value of safety factor applied to cohesion and friction is eventually identical for all the blocks. Alternatively, the magnitude of:the r e q u i ~ slope support force (for example, rock bolts provided at a given orientation) can also be determined by trial and error if a desired value of factor of safety has to be achieved. For polygonal sliding surfaces involving potential wedge sliding, a kinematic stabi!ity analysis for the entire system is first conducted to examine the mode of failure of individual blocks and to determine the contact planes

ROCK MECHANICS IN THE 1990s between blocks. This may be carried out by following the automated procedure as given by Leung and Kheok [5]. If wedge sliding has been identified, the above analytical procedure can still be used to evaluate the stability of the multiple-block sliding surfaces by replacing the orientation of the sliding plane by the relevant line of interception between two adjacent sliding planes. In addition, the resisting force contributed by cohesion would consist of the cohesive forces that can be mobilized along each of the two planes. COMPUTER PROGRAM One major advantage of the vectorial method over the stereographic projection technique is that the entire analysis can be computer coded and automated. In order to achieve this, a computer program has been developed in BASIC programming language to be run on a microcomputer which is currently readily available for design office usage. The structural chart of the program is given in Figure 2.

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determination of minimum bolt force required and the associated bolt orientation to stabilize an excavation, and determination of bolt force required for bolts having the shortest length. The program may also be used to analyze slopes that are protected by sheet pile walls and ground anchors, The passive force resulting from the Coulomb wedge and the tension in the ground anchors can be incorporated in the data file as additional support forces for a particular block. An example problem comprising a system of four free blocks has been employed to illustrate the accuracy of the computer program. The system and its configuration and properties are shown in Figure 3. The stereographic projection plot and the associated force diagram are shown in Figure 4. The graphical method employed basically followed the procedures provided by Goodman [3] for two-block sliding which was extended to four-blocks in the present example in a manner described in the preceding discussion. The results obtained from the graphical method as well as those from the computer program are given in Table 1. Both methods indicate that the system of blocks is highly unstable with a very low factor of safety of 0.31.

Data input by file or on screen Corrections of input data Kinematic analysis and determination of mode of failure

..... cxz slloing plane

---

Int

/f) /

Options for slope stabilization calculations Input desired value of factor of safety 1. 2. 3. 4. 5.

Find Find Find Find Find

permissible range of bolt bearings bolt force required of any given bolt orientation minimum bolt force for a given bolt bearing bolt force for bolt having shortest length cohesive fe-~e required for limit equilibrium Screen & file output

STOP

Fig. 2. Structural chart of computer program The program consists of user-friendly features such as data file input and provision for correction of input data for re-run etc, with the aim that the program may be used by technician personnel for routine and duplicate analyses. Various program options for slope stabilization are also available. These include the automatic determination of factor of safety for a system of blocks, determination of required bolt force of a given orientation to achieve a desired value of factor of safety against sliding, 30:7-EE

oooot \ 13

J ]/

dip angte

t\ ' /

~-20")c" uuuu, -

~

f;.~'\../'~9o/3o ¢ ~-/,D*

For block n Obtai n interaction force from block (n-l) Find net resultant disturbing & resisting force Determine overall stability of system of blocks Determine factor of safety by trial and error

/~J

070/83/(

For block ! to block n-1 Obtain interaction force from last block (except block 1) Find net resultant disturbing & resisting force Determine block is active or passive Determine interaction force for an active block Transfer interaction force to next block

¢:15' .~(~.l~0t/0ip direction/

050L70

z

~:Lu

/o6o/6o

~--3o0 c:o fo; all planes

:30"

Fig. 3. Configuration and properties of example problem In order to stabilize the system to a state of limiting equilibrium (i.e to increase the value of F to 1 such that the resultant force of block 4, R4, would just lie at the fringe of the friction cone on plane 4), one or more of the blocks was reinforced by supporting elements. As an illustrative example, rock bolts were installed at block 4 to stabilize the slope. The range of bearings that rock bolts could be effective for slope stabilization were first determined, as shown in Figure 4 and Table 1. Then the required bolt force at any given bolt orientation, minimum bolt force required and the associated bolt orientation, bolt force required for bolts having shortest length and the associated bolt orientation were subsequently determined graphically and by computer program, as given in Table 1. In the example, no cohesion was assumed to have developed at the sliding planes. If indeed cohesion could be relied upon to stabilize the slope, the magnitude of total cohesive force required at plane 4 in order to achieve limiting equilibrium was determined to be 12,085t by the

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ROCK MECHANICS IN THE 1990s graphical method and 12,001t by the computer program. As Table I shows that the two setsof resultsfor all the cases are similar, the validity of the vectorial method for stability analysis with polygonal sliding surfaces may thereby be reasonably assumed. CASE

STUDY

077 °

0

N

D R

B B. B. C,

0~

Normal or plane Disturbing force Resubon~ force

Belt fro,re at any glvea bearing Minimum bolt force Force for bolts with shortest bolt length Cohesion force required for stabilRy

f

Fig. 4. Stereographic plot and force diagram for example problem

One of the main objectives of developing the vectorial method was to investigate the slope stability of a field case involving a deep basement construction in the weathered sedimentary rocks of Singapore, Although preliminary design using conventional slip analysis by Janbu and Bishop methods indicated that the slope excavations would be safe, several slips occurred during construction aRer prolonged heavy rainfalls. Four slips were observed at the site, as shown in Figure 5. The slope configurations and protection measures of the four collapsed Sections W-W, X-X, Y-Y and Z.Z are shown in Figure 6. A post mortem was conducted using the vectorial method for which the external sliding planes and internal shear planes were oriented along the weakness planes obtained from an earlier geological investigation as well as observed slip surfaces at the site [8]. Accordingly, Section W-W consisted of a single planar sliding surface while the other three sections consisted of polygonal plane sliding surfaces composed of two or three free blocks. In addition, Sections W-W and X-X were found to fail along the weathered shale stratum while the remaining two sections failed along the weathered sandstone and shale strata. ~u CoUapsed j vv /section i /

• ,

"\~

Table 1 Output of example problem

v

Options

Stereonet

Vectorial method

Stability

Unstable

Unstable

Factor of safetyagainst slidingfailure

0.31

Gradient

(F = I)

Permissible range of bolt bearings

77.00 to 123.00

76.90 to 123.10

Bolt force (Be) at trend/plunge 100°/-20°

13,662 t

13,568 t

Minimum bolt foxce (13.) trend 100 ° (nssc~iatedbolt plunge)

10,466 t

10,393 t

(20 °)

(20 e)

Bolt force for rock bolt of shortest length (13,)

20,932 t

20,787 t

Cohesion required (C,)

12,085 t

12,001 t

/z

v, x-li' 'x

0.31

Limit equilibriumanalysis

•_

---

1:o.9

--

yi

--

Fig. 5. Site plan showing collapsed slope sections A back-analysis was performed in order to establish the strength ~ n m t e r s of the rock m s , The coh~ion an.d frictionangles of the weathered shale were found to be within a range of 20 kPa to 2 5 k P a and 20 ° to 25,0 respectively. As for the w e a t b e ~ sandstone, the cohesion was 6 kPa and the friction angle 32 °. These

ROCK MECHANICS IN THE 1990s properties indicated that the shale had probably softened and contained a small amount of clayey materials. This tendency was substantiated by the fact that the shales were found from site investigation, to contain expansive materials like sericite-monotmorillonite. Furthermore, records of ground water table observations indicated fluctuations of the ground water level that were likely to have caused repeated shrinkage of shales due to drying followed by swelling due to wetting by a rising water table, resulting in decomposition of the shale. Notes: (1) 1, 2, 3 refer to sliding planes 1, 2, 3, respectively (2) 12, 23 refer Io shear planes 12, 23, respectively

[

Shotclete

o~ ~

Shale

((1) Section W-W

1583

Remedial works were implemented at the site after the collapse of the slopes. The slope protection measures included laying shotcrete on the slope surface, and installing rock bolts, ground anchors and sheet piles (see Figure 7). Further analyses were carried out using the vectorial approach to evaluate the slope stabilities. The potential failure surfaces were assumed to pass through the toes of the sheet piles, causing deep-seated failures and the weakness planes were taken as internal shear surfaces. By trial and error, the admissible failure surfaces which give the minimum safety factor are shown in Figure 7. As mentioned earlier, the passive force resulting from Coulomb wedges and the tension in the ground anchors were input as additional support forces to the relevant blocks. The values of safety factor obtained are also given in Table 2. Although all the values are well above one, it may be noted that the margin for safety for slope Section X-X was rather low. This may be attributed to the fact the ground anchor bond length used was only 4.8 m which was considerably shorter than the bond length of 6.5 m for the other slopes.

(b) Section X-X

Sh0tcrete --

W.I.

-

(el Section Y-Y

/

o w.T,

w.,.

Idl Section l-I _

(o) Section W-W

(b) Section X-X

Fig. 6. Side elevation of slope sections For these strength parameters, the corresponding factors of safety obtained using the proposed vectorial approach are shown in Table 2. The values may be noted to be about unity varying within a small margin between 0.97 and 1.05. Conventional slope stability analyses using Janbu's [9] composite slip surface approach were also carried out to determine the factor of safety of the collapsed sections and to compare the results with those obtained using the vectorial method. Based on identical slope configurations and strength parameters, the values of factor of safety of the four slope sections are also given in Table 2. As the values were found to vary substantially from 0.87 to 1.34, it is apparent that the conventional slope stability approach may not be appropriate for the analysis of such weathered rock slopes.

--

W.T.

:i.i./ (c) Section Y-Y

(d) Section Z-Z

Fig. 7. Side elevation of slope sections aider remedial works

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ROCK MECHANICS IN THE 1990s Table 2 Values of fact.ors of safety Failed

slopes

After

remedial work

Section

Janbu

Vectorial

Janbu

Vectorial

method

method

method

method

W-W

1.34

1.1M

1.44

1.77

X-X

1.06

1.05

0.95

1.15

Y-Y

0.87

0.97

1.29

1.34

Z-Z

0.89

0.98

1.45

1.74

and run on micro-computers which are readily available in design practice. The validity of the proposed method has been verified against the results obtained via the stereographic projection technique. The method has also been applied in the back-analysis of a field case study involving slope excavations in weathered sedimentary rocks. The results show that the proposed method is more applicable for the analysis of weathered rock slopes while conventional slope stability methods such as Janbu and Bishop methods may not be amenable to the analysis of such slopes. REFERENCES I.

Janbu method was also employed to back-analyze the stability of the slopes after remedial works and the results are also given in Table 2. The values of safety factor obtained from Janbu method are significantly different from those obtained using the vectorial method. As the earlier analyses for collapsed slope sections suggest that the vectorial method is more appropriate for the stability analysis of weathered rock slopes, the discrepancies in the value of safety factor once again indicate that conventional soil slope stability methods may give misleading outcome for the stability evaluation of such slopes. CONCLUSION The relatively simple vectorial method has been developed to tackle the stability problem of multiple-block sliding surfaces. The entire analysis has been automated

Hock E. and Bray J. Rock Slope Engineering (third edition). Institutionof Mining and Metallurgy, London (1981). 2. Kovari K. and Fritz P: Slope stabilitywith plane, wedge and polygonal slidingsurfaces.Proc. Int. ~ymp. on Rock Mech. Related to D a m Foundations, Rio de Janerio(1978). 3. Goodman R. E. Methods of Geological Engineering in Discontinuous Rocks. West Pub. Co., St. Paul (1976). 4. Goodman R. E. and Shi G. Block Theory and Its Applications to Rock Engineering. Prentice-Hail, Englewood Cliffs,N.J. (1985). 5. Lcung C. F. and Kheok S. C. Computer aided analysisof rock slopestability.Rock Mech. and Rock Engineering,20, 111-122 (1987). 6. Leung C. F. Computer aided design of underground excavations in jointed rock. Rock Mech. and Rock Engineering, 23, 71-89 (1990) 7. Goodman R.E. Introduction to Rock Mechanics (second edition).John Wiley and Sons, N e w York (1989). 8. Lo K. W. Leung C. F. Hayata K, and Lee S.L. Stability of excavated slopes in the weathered Jurong Formation of Singapore. Proc 2nd Int. Conf. on Geomechanics in Tropical Soils, I, Singapore, 277-284 (1988). 9. Janbu N. Applicationsof composite slipcirclesfor stability analysis. Proc. European Conf. on Stabilityof Earth Slopes, 3, Stockholm, 43-49 (1954)o